Thursday, March 17, 2011 - 3:00pm
1 hour (actually 50 minutes)
Grothendieck's anabelian conjectures say that hyperbolic curves over certain fields should be K(pi,1)'s in algebraic geometry. It follows that points on such a curve are conjecturally the sections of etale pi_1 of the structure map. These conjectures are analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. This talk will start with an introduction to Grothendieck's anabelian conjectures, and then present a 2-nilpotent real section conjecture: for a smooth curve X over R with negative Euler characteristic, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that the set of real points equipped with a real tangent direction of the smooth compactification of X is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(R) action, showing a 2-nilpotent birational real section conjecture.